Problem 1565

1. First Move:
Terence's first move is rock. This is given and has a probability of 1.

2. Second Move:
On the second move, Terence will play either paper or scissors, each with a probability of $\frac{1}{2}$.

3. Third Move:
We now consider two cases for the third move:
- Case 1: The second move is paper.
- The third move will be either rock or scissors, each with a probability of $\frac{1}{2}$.
- Case 2: The second move is scissors.
- The third move will be either rock or paper, each with a probability of $\frac{1}{2}$.

4. Fourth Move:
We now consider the fourth move for each sub-case of the third move:
- Case 1.1: The second move is paper, and the third move is rock.
- The fourth move will be either paper or scissors, each with a probability of $\frac{1}{2}$.
- Case 1.2: The second move is paper, and the third move is scissors.
- The fourth move will be either rock or paper, each with a probability of $\frac{1}{2}$.
- Case 2.1: The second move is scissors, and the third move is rock.
- The fourth move will be either paper or scissors, each with a probability of $\frac{1}{2}$.
- Case 2.2: The second move is scissors, and the third move is paper.
- The fourth move will be either rock or scissors, each with a probability of $\frac{1}{2}$.

5. Fifth Move:
We now consider the fifth move for each sub-case of the fourth move:
- Case 1.1.1: The second move is paper, the third move is rock, and the fourth move is paper.
- The fifth move will be rock with a probability of $\frac{1}{2}$.
- Case 1.1.2: The second move is paper, the third move is rock, and the fourth move is scissors.
- The fifth move will be rock with a probability of $\frac{1}{2}$.
- Case 1.2.1: The second move is paper, the third move is scissors, and the fourth move is rock.
- The fifth move cannot be rock.
- Case 1.2.2: The second move is paper, the third move is scissors, and the fourth move is paper.
- The fifth move will be rock with a probability of $\frac{1}{2}$.
- Case 2.1.1: The second move is scissors, the third move is rock, and the fourth move is paper.
- The fifth move will be rock with a probability of $\frac{1}{2}$.
- Case 2.1.2: The second move is scissors, the third move is rock, and the fourth move is scissors.
- The fifth move will be rock with a probability of $\frac{1}{2}$.
- Case 2.2.1: The second move is scissors, the third move is paper, and the fourth move is rock.
- The fifth move cannot be rock.
- Case 2.2.2: The second move is scissors, the third move is paper, and the fourth move is scissors.
- The fifth move will be rock with a probability of $\frac{1}{2}$.

6. Combining Probabilities:
- For Case 1.1: $\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$
- Probability of fifth move being rock: $\frac{1}{4} \times \frac{1}{2} = \frac{1}{8}$
- For Case 1.2: $\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$
- Probability of fifth move being rock: $\frac{1}{4} \times \frac{1}{2} = \frac{1}{8}$
- For Case 2.1: $\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$
- Probability of fifth move being rock: $\frac{1}{4} \times \frac{1}{2} = \frac{1}{8}$
- For Case 2.2: $\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$
- Probability of fifth move being rock: $\frac{1}{4} \times \frac{1}{2} = \frac{1}{8}$

7. Total Probability:
$$\frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8} = \frac{4}{8} = \frac{1}{2}$$

The final answer is $\boxed{\frac{1}{2}}$.